Compute int_{0}^{ln (2) / 4} frac{e^{4 x}}{left(e^{4 x}+4right)^{7}} d x

Question

Accepted Answer

Step:1 ：Let ( u = e^{4x} + 4 ), then ( du = 4e^{4x}dx )Step:2 ：Change the limits of integration to match the substitution: lower limit ( u(0) = 5 ), upper limit ( u(ln(2)/4) = 2^{1/4} + 4 )Step:3 ：Rewrite the integral in terms of ( u ) and ( du ): ( int frac{1}{4u^{7}} du )Step:4 ：Integrate to find ( -frac{5}{4u^{6}} ) evaluated from ( 5 ) to ( 2^{1/4} + 4 )Step:5 ：Plug in the limits of integration: ( -frac{5}{4(2^{1/4} + 4)^{6}} + frac{5}{4(5)^{6}} )Step:6 ：Simplify the expressionStep:7 ：( boxed{-frac{5}{4 left(e^{4 x}+4right)^{6}}+frac{2^{1 / 4}+4}{4 left(e^{4 x}+4right)^{6}}} )

Answer

Step: 1 ：Let ( u = e^{4x} + 4 ), then ( du = 4e^{4x}dx )Step: 2 ：Change the limits of integration to match the substitution: lower limit ( u(0) = 5 ), upper limit ( u(ln(2)/4) = 2^{1/4} + 4 )Step: 3 ：Rewrite the integral in terms of ( u ) and ( du ): ( int frac{1}{4u^{7}} du )Step: 4 ：Integrate to find ( -frac{5}{4u^{6}} ) evaluated from ( 5 ) to ( 2^{1/4} + 4 )Step: 5 ：Plug in the limits of integration: ( -frac{5}{4(2^{1/4} + 4)^{6}} + frac{5}{4(5)^{6}} )Step: 6 ：Simplify the expressionStep: 7 ：( boxed{-frac{5}{4 left(e^{4 x}+4right)^{6}}+frac{2^{1 / 4}+4}{4 left(e^{4 x}+4right)^{6}}} )

Compute $\int_{0}^{\ln (2) / 4} \frac{e^{4 x}}{{(e^{4 x} + 4)}^{7}} d x$

Answer

Popular Problems

Compute ∫0ln⁡(2)/4e4x(e4x+4)7dx

Answer

Popular Problems

Compute $\int_{0}^{\ln (2) / 4} \frac{e^{4 x}}{{(e^{4 x} + 4)}^{7}} d x$